I've provided a detailed explanation
below using State 3 as an example, as well as further notes on State 5 and State
6. Note that these examples are not the only
ways to solve States 3, 5 and 6 using the two basic LL edge orientation algorithms,
they are
merely examples of one way to do it.

Please note: Before you ask,
yes, I know that States 4 and 7 are not on this page. It is deliberate that only some of the LL orientations are explained to the nth degree.
This is to encourage people to do some thinking and not just blindly apply algorithms without
paying any attention to what the algorithms do! I am still very happy for people
to email me with questions, but please have a go at it yourself first. You will
find it more rewarding if you discover things yourself! :)

Orienting the LL Corners

There are 8 possible orientation states for the LL corners. One is where all 4 corners are correctly oriented. The other 7 look like this.

State 1

State 2

State 3

State 4

State 5

State 6

State 7

State 1. Twisting three corners anti-clockwise

R' U' R U' R' U2 R U2

Instead of just blindly executing this algorithm, think about what it is
doing to the LL corners. This algorithm is spinning 3 LL corners 120 degrees
anti-clockwise (corners marked BR, FR and FL), and leaving the orientation of one LL corner unaffected
(corner BL).

State 2. Twisting three corners clockwise

R U R' U R U2 R' U2

As with the State 1 algorithm, think about what the State 2 algorithm is
actually doing. The State 2 algorithm is spinning 3 LL corners 120 degrees
clockwise (corners marked BL, BR and FR), and leaving the orientation of one LL corner unaffected
(corner FL).

From thinking about what the State 2 algorithm is actually doing to the cube, we
know that it will spin three of the LL corners 120 degrees clockwise (corners
BL, BR, FR), and leave one corner unaffected (corner FL). This means that:

the BL corner and the BR corner (which have correct orientation in the
picture above) will have the incorrect orientation after this step; and

the FR corner will have the correct orientation after this step; and

the FL corner will remain with the incorrect orientation after this
step since it was unaffected by the State 2 algorithm performed during this
step.

So what you're left with after this step are 3 incorrectly oriented LL corners,
and 1 correctly oriented LL corner. Now take a look at 'State 3. Second step'.

State 3. Second step

Now take a close look at what you've
got. One of the LL corners is correctly oriented, and 3 of the LL corners are
incorrectly oriented.

The 3 corners which are incorrectly oriented all need to be spun 120 degrees
anti-clockwise. Does this sound familiar? Take a look at the description above
for the State 1 algorithm: "Twisting three corners anti-clockwise".

If
you now do a U2 turn (which basically just means you are looking at your U layer
from a different angle), you can then perform the State 1 algorithm R' U' R U' R' U2 R U2.
This will complete the LL corner orientation.

Now re-examine the orientation of your
last layer corners. One of the LL corners is correctly oriented, and 3 of the LL corners are
incorrectly oriented.

The 3 corners which are incorrectly oriented all need to be spun 120 degrees
clockwise. If you do a U turn, your LL will look the same as State 2,
which means you can then perform the State 2 algorithm R U R' U R U2 R' U2.
This will complete the LL corner orientation.

Now re-examine the orientation of your
last layer corners. One of the LL corners is correctly oriented, and 3 of the LL corners are
incorrectly oriented.

The 3 corners which are incorrectly oriented all need to be spun 120 degrees
anti-clockwise. If you do a U' turn, your LL will look the same as State 1,
which means you can then perform the State 1 algorithm R' U' R U' R' U2 R U2.
This will complete the LL corner orientation.

If you are still having trouble with orienting the LL corners, have a look at
the third
layer section of Ryan Heise's beginner solution - go to the part called
'Make the corners face up'.